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First-Order LogicChapter 8

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Outline

Why FOL? Syntax and semantics of FOL

Using FOL Wumpus world in FOL Knowledge engineering in FOL

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Pros and cons of propositional

logic Propositional logic is declarative Propositional logic allows partial/disjunctive/negated

information (unlike most data structures and databases)

Propositional logic is compositional:

meaning ofB1,1 P1,2 is derived from meaning ofB1,1 and ofP1,2

Meaning in propositional logic is context-independent

(unlike natural language, where meaning depends on context)

Propositional logic has very limited expressive power

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First-order logic

Whereas propositional logic assumes theworld contains facts,

first-order logic (like natural language)assumes the world contains

Objects: people, houses, numbers, colors,

baseball games, wars, Relations: red, round, prime, brother of,

bigger than, part of, comes between,

Functions: father of, best friend, one more

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Syntax of FOL: Basic elements

Constants KingJohn, 2, NUS,... Predicates Brother, >,...

Functions Sqrt, LeftLegOf,... Variables x, y, a, b,... Connectives , , , ,

Equality = Quantifiers ,

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Atomic sentences

Atomic sentence = predicate (term1,...,termn)orterm1 = term2

Term = function (term1,...,termn)orconstantorvariable

E.g., Brother(KingJohn,RichardTheLionheart) >

(Length(LeftLegOf(Richard)),Length(LeftLegOf(KingJohn)))

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Complex sentences

Complex sentences are made from atomicsentences using connectives

S, S1 S2, S1 S2, S1 S2, S1 S2,

E.g. Sibling(KingJohn,Richard) Sibling(Richard,KingJohn)>(1,2) (1,2)>(1,2) >(1,2)

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Truth in first-order logic Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements) and relations amongthem

Interpretation specifies referents forconstant symbols objects

predicate symbols relations

function symbols functional relations

An atomic sentencepredicate(term1,...,termn) is trueiff the objects referred to by term1,...,termnare in the relation referred to bypredicate

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Models for FOL: Example

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Universal quantification

Everyone at NUS is smart:x At(x,NUS) Smart(x)

x P is true in a model m iffP is true with xbeing eachpossible object in the model

Roughly speaking, equivalent to the conjunction of

instantiations ofP

At(KingJohn,NUS) Smart(KingJohn)

At(Richard,NUS) Smart(Richard)

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A common mistake to avoid

Typically, is the main connective with

Common mistake: using

as the mainconnective with :x At(x,NUS) Smart(x)means Everyone is at NUS and everyone is smart

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Existential quantification

Someone at NUS is smart: xAt(x,NUS) Smart(x)$

x P is true in a model m iffP is true with xbeing some

possible object in the model

Roughly speaking, equivalent to the disjunction ofinstantiations ofP

At(KingJohn,NUS) Smart(KingJohn)

At(Richard,NUS) Smart(Richard)

At(NUS,NUS) Smart(NUS)

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Another common mistake to

avoid Typically, is the main connective with Common mistake: using as the main

connective with :

xAt(x,NUS) Smart(x)

is true if there is anyone who is not at NUS!

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Properties of quantifiers x y is the same as y x x y is the same as y x

x y is not the same as y x x y Loves(x,y)

There is a person who loves everyone in the world

y

x Loves(x,y) Everyone in the world is loved by at least one person

Quantifier duality: each can be expressed using the other

x Likes(x,IceCream)

x

Likes(x,IceCream)

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Equality

term1 = term2 is true under a given interpretationif and only ifterm1 and term2 refer to the sameobject

E.g., definition ofSiblingin terms ofParent:

x,ySibling(x,y) [(x = y) m,f (m = f) Parent(m,x) Parent(f,x) Parent(m,y) Parent(f,y)]

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Using FOL

The kinship domain:

Brothers are siblings

x,y Brother(x,y) Sibling(x,y)

One's mother is one's female parent

m,c Mother(c) = m (Female(m) Parent(m,c))

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Using FOL

The set domain:

s Set(s) (s = {} ) (x,s2 Set(s2) s = {x|s2})

x,s {x|s} = {} x,s x s s = {x|s} x,s x s [y,s2} (s = {y|s2} (x = y x s2))] s1,s2 s1 s2 (x x s1 x s2) s1,s2 (s1 = s2) (s1 s2 s2 s1)

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Interacting with FOL KBs Suppose a wumpus-world agent is using an FOL KB and perceives a smell

and a breeze (but no glitter) at t=5:

Tell(KB,Percept([Smell,Breeze,None],5))Ask(KB,a BestAction(a,5))

I.e., does the KB entail some best action at t=5?

Answer: Yes, {a/Shoot} substitution (binding list)

Given a sentence Sand a substitution , S denotes the result of plugging into S; e.g.,

S= Smarter(x,y) = {x/Hillary,y/Bill}S = Smarter(Hillary,Bill)

Ask KB,S returns some/all such that KB

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Knowledge base for the

wumpus world Perception t,s,b Percept([s,b,Glitter],t) Glitter(t)

Reflex t Glitter(t) BestAction(Grab,t)

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Deducing hidden properties

x,y,a,b Adjacent([x,y],[a,b]) [a,b] {[x+1,y], [x-1,y],[x,y+1],[x,y-1]}

Properties of squares: s,t At(Agent,s,t) Breeze(t) Breezy(s)

Squares are breezy near a pit:

Diagnostic rule---infer cause from effects Breezy(s) \Exi{r} Adjacent(r,s) Pit(r)$

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Knowledge engineering in FOL

1. Identify the task2.2. Assemble the relevant knowledge

3.3. Decide on a vocabulary of predicates,

functions, and constants4.4. Encode general knowledge about the domain5.5. Encode a description of the specific problem

instance

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The electronic circuits domain

One-bit full adder

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1. Identify the task2.

Does the circuit actually add properly? (circuitverification)

2. Assemble the relevant knowledge3.

Composed of wires and gates; Types of gates (AND,

OR, XOR, NOT) Irrelevant: size, shape, color, cost of gates

3. Decide on a vocabulary

4.

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4. Encode general knowledge of the domain5.

t1,t2 Connected(t1, t2) Signal(t1) = Signal(t2)

t Signal(t) = 1

Signal(t) = 0 1 0 t1,t2 Connected(t1, t2) Connected(t2, t1)

g Type(g) = OR Signal(Out(1,g)) = 1 nSignal(In(n,g)) = 1

g Type(g) = AND Signal(Out(1,g)) = 0 n

Signal(In(n,g)) = 0

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5. Encode the specific problem instance6.

Type(X1) = XOR Type(X2) = XORType(A1) = AND Type(A2) = ANDType(O1) = OR

Connected(Out(1,X1),In(1,X2)) Connected(In(1,C1),In(1,X1))Connected(Out(1,X1),In(2,A2)) Connected(In(1,C1),In(1,A1))Connected(Out(1,A2),In(1,O1)) Connected(In(2,C1),In(2,X1))

Connected(Out(1,A1),In(2,O1)) Connected(In(2,C1),In(2,A1))Connected(Out(1,X2),Out(1,C1)) Connected(In(3,C1),In(2,X2))Connected(Out(1,O1),Out(2,C1)) Connected(In(3,C1),In(1,A2))

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6. Pose queries to the inference procedure7.

What are the possible sets of values of all theterminals for the adder circuit?

i1,i2,i3,o1,o2 Signal(In(1,C_1)) = i1 Signal(In(2,C1)) =

i2 Signal(In(3,C1)) = i3 Signal(Out(1,C1)) = o1 Signal(Out(2,C1)) = o2

7. Debug the knowledge base

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Summary

First-order logic:

objects and relations are semantic primitives syntax: constants, functions, predicates,equality, quantifiers

Increased expressive power: sufficient todefine wumpus world

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